Lagrangian flows driven by $BV$ fields in Wiener spaces

TL;DR

This paper proves the renormalization property for bounded solutions of the continuity equation with BV vector fields in Wiener spaces, leading to new uniqueness and stability results for associated Lagrangian flows, with an example on elliptic problems.

Contribution

It establishes the renormalization property for BV fields in Wiener spaces and derives new uniqueness and stability results for Lagrangian flows, extending classical results to infinite-dimensional settings.

Findings

Renormalization property for BV fields in Wiener spaces proven.

New uniqueness results for Lagrangian flows established.

Stability results for solutions demonstrated.

Abstract

We establish the renormalization property for essentially bounded solutions of the continuity equation associated to $BV$ fields in Wiener spaces, with values in the associated Cameron-Martin space; thus obtaining, by standard arguments, new uniqueness and stability results for correspondent Lagrangian $L^\infty$-flows. An example related to Neumann elliptic problems is also discussed.